This introduction is written with Alpha in mind, and will almost certainly change as we move into beta and release.
Most characters in Arx are a member of one or more organizations, that represent just how they are plugged into the social matrix of Arx. A noble vassal would likely be a member of their own House, the great House that they owe fealty to, and then public institutitions like the Faith of the Pantheon or groups such as the Champions guild. Tasks are an abstract means of showing a character accumulating contacts, influence and resources while acting on behalf of these organizations, to benefit themselves and the organizations they represent.
Every organization has a number of example tasks available that represent thematic elements unique to the group. Tasks are divided into three categories: social tasks, economic tasks, and military tasks. They are viewable with '+task <organization name>', such as a noble in house Grayson typing '+task Grayson'. When a character finds a task that fits their character as a goal that they would pursue, they accept the task, then attempt to convince others in role-play to support their goals, and use the 'task/supportme' command for others to register their support. Then, at the end of the week during the weekly update, if a task has sufficient support it will complete and reward both the character completing the task and the organization they repersented with an equal amount of silver and resources that match the task type (social resources for a social task, for example.) In addition the character will receive positive reputation with the organization they worked for, and potentially negative reputation with the organization's enemies. If a task does not complete, it stays active until it reaches enough support to complete or abandoned, and counts against one of the 7 weekly tasks that a character can complete per week.
For example, Princess Exempla of House Grayson uses '+task Grayson' and sees task #2, called 'Stiff Upper Lip', which is asking nobles to try to help stem the tide of hysteria in the city that has been caused by the most recent crisis, and to try to recruit others to keep panic in check. She meets Prince Phil, Lord Fred, and Guardsman Bob, and roleplays with them about this task, arguing about the importance of a calm and collected approach, and then asks if they will support her, and uses the '+task/supportme 2,Grayson=phil,fred,bob'. That command represents that she is asking Phil, Fred, and Bob for support on task #2, 'Stiff Upper Lip', on behalf of House Grayson.
In character, what she is asking is for Phil, Fred, and Bob to use their own influence to stem the tide of panic. Each character has their own weekly 'support' total that they can use to donate to tasks, to represent the amount of influence they have in the city and how they use it. Prince Phil's monocle drops into his tea in horror at Exempla's pleas, and uses his full support total for the week to back her. As an abstraction, the support total represents him being true to his word and immediately flouncing back to all his vassals and agreeing something must be done, and rallying his forces to her cause. It is also completely hidden from the person he is supporting, so Exempla only has Phil's good word and the fact that her task will complete at the end of the week to know this. For comparison, Guardsman Bob is sympathetic, but he would much rather support the courtier he is fond of that was asking to meet him later to plea for his help in solving a dilemma, so he agrees but only gives a default 1 support, which does not cost against his weekly total. Lord Fred, on the other hand, is actually secretly responsible for much of the unrest and is plotting to destroy Exempla and all she stands for, so he makes many sympathetic words, agrees to help, but gives no support at all and instead plans to tell all of Exempla's enemies what she is up to. Shame on you, Fred.
Nonetheless, at the end of the week, Exempla's task, 'Stiff Upper Lip', a difficulty 15 task, passed the 15 support threshold to complete. As a representation of the good this caused, for her and house Grayson, she earns silver (probably from grateful vassals of House Grayson saying sales are picking up again due to reduced panic) and social resources (those same grateful individuals willing to be her eyes and ears). She can in turn spend these resources in Dominion, for agents and other systems we still have yet to define.
Every week, a character can do up to 7 tasks, and the 7 refresh each week unless a task stays active and incomplete. As a way to use up extraneous tasks, a character can use the '+task/work' command which uses up a task for the week and returns a minor amount of silver and a single resource of the given type. This is supposed to be a very slight opportunity cost to tasks, but still intended to be an active rather than passive source of income.
Tasks will likely change more as we go into beta and then release, feel free to give feedback.